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| Description
| - A simple topological graph T=(V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2^{O(n^2 log (m/n))}, and at most 2^{O(mn^{1/2} log n)} if m<n^{3/2}. As a consequence we obtain a new upper bound 2^{O(n^{3/2} log n)} on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2^{n^2 alpha(n)^{O(1)}}, using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2^{m^2+O(mn)} and at least 2^{Omega(m^2)} for graphs with m>(6+epsilon)n.
- A simple topological graph T=(V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2^{O(n^2 log (m/n))}, and at most 2^{O(mn^{1/2} log n)} if m<n^{3/2}. As a consequence we obtain a new upper bound 2^{O(n^{3/2} log n)} on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2^{n^2 alpha(n)^{O(1)}}, using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2^{m^2+O(mn)} and at least 2^{Omega(m^2)} for graphs with m>(6+epsilon)n. (en)
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| Title
| - Improved enumeration of simple topological graphs
- Improved enumeration of simple topological graphs (en)
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| skos:prefLabel
| - Improved enumeration of simple topological graphs
- Improved enumeration of simple topological graphs (en)
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| skos:notation
| - RIV/00216208:11320/13:10145590!RIV14-GA0-11320___
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| http://linked.open...avai/riv/aktivita
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| http://linked.open...avai/riv/aktivity
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| http://linked.open...iv/cisloPeriodika
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| http://linked.open...vai/riv/dodaniDat
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| http://linked.open...aciTvurceVysledku
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| http://linked.open.../riv/druhVysledku
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| http://linked.open...iv/duvernostUdaju
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| http://linked.open...titaPredkladatele
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| http://linked.open...dnocenehoVysledku
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| http://linked.open...ai/riv/idVysledku
| - RIV/00216208:11320/13:10145590
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| http://linked.open...riv/jazykVysledku
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| http://linked.open.../riv/klicovaSlova
| - Isomorphism of topological graphs; Weak isomorphism of topological graphs; Simple topological graph; Simple complete topological graph (en)
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| http://linked.open.../riv/klicoveSlovo
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| http://linked.open...odStatuVydavatele
| - US - Spojené státy americké
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| http://linked.open...ontrolniKodProRIV
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| http://linked.open...i/riv/nazevZdroje
| - Discrete and Computational Geometry
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| http://linked.open...in/vavai/riv/obor
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| http://linked.open...ichTvurcuVysledku
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| http://linked.open...cetTvurcuVysledku
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| http://linked.open...vavai/riv/projekt
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| http://linked.open...UplatneniVysledku
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| http://linked.open...v/svazekPeriodika
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| http://linked.open...iv/tvurceVysledku
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| http://linked.open...ain/vavai/riv/wos
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| issn
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| number of pages
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| http://bibframe.org/vocab/doi
| - 10.1007/s00454-013-9535-8
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| http://localhost/t...ganizacniJednotka
| |
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